Appendix A - Lie Groups and Algebras

Summary

We list here some of the basic facts concerning Lie groups and algebras that are needed in this book. Complete proofs and further details can be found in [85]. Since almost all Lie groups that arise in mathematical physics are groups of matrices, we shall confine our attention to local linear Lie groups.

Let W be an open, connected set containing e = (0,…, 0) in the space Rⁿ of all real wn-tuples g = (g₁,…,g_n).

Definition. An n-dimensional (real) local linear Lie group G is a set of m×m nonsingular complex matrices A(g) = A(g₁,…,g_n) defined for each g∈ W such that

1. A (e) = E_m (the identity matrix).

2. The matrix elements of A (g) are analytic functions of the parameters g₁,…,g_n and the map g→A(g) is one to one.

3. The n matrices δA(g)/δg_j,j= 1,…,n, are linearly independent for each g∈W.

4. There exists a neighborhood W^t of e in Rⁿ, W′ ⊂ W, with the property that for every pair of n-tuples g,h in W′ there is an n-tuple k in W satisfying A (g)A (h) = A (k) where the operation on the left is matrix multiplication.